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Gap Sequences of 1-Weierstrass Points on Non-Hyperelliptic Curves

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Gap Sequences of 1-Weierstrass Points on Non-Hyperelliptic Curves Gap sequences of 1-Weierstrass points on non-hyperelliptic curves of genus 10 Mohammed A. Saleem1 and Eslam E. Badr2 1Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt 2Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Emails: [email protected] [email protected] Abstract In this paper, we compute the 1-gap sequences of 1-Weierstrass points of non-hyperelliptic smooth projective curves of genus 10. Furthermore, the ge- ometry of such points is classified as flexes, sextactic and tentactic points. Also, an upper bounds for their numbers are estimated. MSC 2010: 14H55, 14Q05 Keywords: 1-Weierstrass points, q-gap sequence, flexes , sextactic points, tentactic points, kanonical linear system, , Kuribayashi sextic curve. arXiv:1307.0078v1 [math.AG] 29 Jun 2013 1 0 Introduction Weierstrass points on curves have been extensively studied, in connection with many problems. For example, the moduli space Mg has been stratified with subvarieties whose points are isomorphism classes of curves with particular Weierstrass points. For more deatails, we refer for example [3], [6]. At first, the theory of the Weierstrass points was developed only for smooth curves, and for their canonical divisors. In the last years, starting from some papers by R. Lax and C. Widland (see [10], [11], [12], [13], [14], [18]), the theory has been reformulated for Gorenstein curves, where the invertible dualizing sheaf substitutes the canonical sheaf. In this contest, the singular points of a Gorenstein curve are always Weierstrass points. In [16], R. Notari developed a technique to compute the Weierstrass gap sequence at a given point, no matter if it is simple or singular, on a plane curve, with respect 0 to any linear system V ⊆ H C,OC(n) . This technique can be useful to construct examples of curves with Weierstrass points of given weight, or to look for conditions for a sequence to be a Weierstrass gap Sequence. He used this technique to compute the Weierstrass gap sequence at a point of particular curves and of families of quintic curves. In this paper, we compute the 1-gap sequence of the 1-Weierstrass points on smooth non-hyperelliptic algebraic curves of genus 10 which can be embedded holo- morphically and effectively in algebraic curves of degree 6. Furthermore, the geome- try of Weierstrass points is classified as flexes, sextactic and tentactic points. On the other hand, we show that a smooth non-hyperelliptic curve of genus 10 has no 4-flex, 1, 2-sextactic or 9-tentactic points. Also, an upper bound of the numbers of flexes, sextactic and tentactic points on such curves is estimated. 1 Preliminaries Notations. We assume the following notations throughout the present paper. I(C1,C2; p); the intersection number of the curves C1 and C2 at the point p [4]. (q) Gp (Q); the q-gap sequence of the point p with respect to the linear system Q [1, 2]. ω(q)(p); the q-weight of the point p [15]. N (q)(C); the number of q-Weierstrass points on C [8]. 2 Q − (ℓ.p) ; the set of divisors in the linear system Q with multiplicity at least ℓ at the point p [15]. Lemma 1.1. [9, 15] Let X be a smooth projective plane curve of genus g. The number of q-Weierstrass points N (q)(C), counted with their q-weights, is given by g(g2 − 1), if q =1 (q) N (C)= (2q − 1)2(g − 1)2g, if q ≥ 2. In particular, for smooth projective plane sextic (i.e. g = 10), the number of 1- Weierstrass points is 990 counted with their weights. (1) Theorem 1.2. [15] Let X be a non-hyperelliptic curve of genus ≥ 3. Write Gp (Q)= {n1 < n2 < ... < ng}, then (1) n1 =1, (2) nr ≤ 2r − 2 for every r ≥ 2, (g − 1)(g − 2) (3) ω(1)(p) ≤ . 2 (4) There are at least 2g + 6 1-Weierstrass points on X. Remark 1.3. For more details on q-Weierstrass points on Riemann surfaces, we refer for example to [15, 5]. 2 Main results Let X be a smooth projective plane curve of genus 10, and let Q := |K| be its canonical linear system. 9 Proposition 2.1. The linear system Q is g18. Proof. The result is an immediate consequence, since dim Q := dim |K| = g − 1=9, and deg (Q) := deg (K)=2(g − 1) = 18. (1) (1) Corollary 2.2. Let p ∈ X, then ♯Gp (Q)=10 and Gp (Q) ⊂{1, 2, 3, ..., 19}. 9 Lemma 2.3. The set of cubic divisors on X form a linear system which is g18. 3 2.1 Flexes Definition 2.4. [1, 17] A point p on a smooth plane curve C is said to be a flex point if the tangent line Lp meets C at p with contact order Ip(C, Lp; p) at least three. We say that p is i-flex, if Ip(C, Lp) − 2= i. The positive integer i is called the flex order of p. Our main results for this part are the following. Theorem 2.5. Let p be a flex point on a smooth projective non-hyperelliptic plane curve C of g = 10. Let Lp be the tangent lint to C at p such that I(C, Ep)= µf . Then (1) Gp (Q)= {1, 2, 3, 1+ µf , 2+ µf , 3+ µf , 2µf +1, 2µf +2, 3µf +1, 3µf +2}. Moreover, the geometry of such points is given by the following table: (1) ωp (Q) Gp(Q) Geometry 2 {1,2,3,...,8,10,11} 1-flex 15 {1,2,3,5,6,7,9,10,13,14} 2-flex 28 {1,2,3,6,7,8,11,12,16,17} 3-flex Proof. The dimensions of Q(−1.p) and Q(−2.p) do not depend on whether p is 1- Weierstrass point or not, i.e., (1) 1, 2 ∈ Gp (Q). The spaces Q(−3.p)= ... = Q(−µf .p), consists of divisor of cubic curves of the form LpR, where R is an arbitrary conic. Hence, dim Q(−ℓ.p) =6 for ℓ = 3, ..., µf . That is, (1) 3 ∈ Gp (Q). The space Q − (1+ µf ).p consists of divisor of cubic curves of the form LpR, where R is a conic passing through p. Hence, dim Q − (1 + µf ).p =5. That is, (1) 1+ µf ∈ Gp (Q). The space Q −(2+µf ).p consists of divisor of cubic curves of the form LpR, where R is a conic passing through p with contact order at least 2. Hence, dim Q −(2+µf ).p = 4. That is, (1) 2+ µf ∈ Gp (Q). The spaces Q − (3 + µf ).p = ... = Q − 2µf .p consists of divisor of cubic curves of the form L2H, where H is an arbitrary hyperplane. Hence, dim Q − ℓ.p = 3 for p ℓ =3+ µf , ..., 2µf . That is, (1) 3+ µf ∈ Gp (Q). 4 2 The space Q −(2µf +1).p consists of divisor of cubic curves of the form L H, where p H is a hyperplane through p. Hence, dim Q − (2µf + 1).p = 2. That is, (1) 2µf +1 ∈ Gp (Q). The spaces Q − (2µf + 2).p = ... = Q − 3µf .p consists of divisor of cubic curves 2 of the form LpH, where H is a hyperplane through p with contact order at least 2. Hence, dim Q − (2µf + 2).p =1. That is, (1) 2µf +2 ∈ Gp (Q). 3 The space Q − (3µf + 1).p consists of divisor of the cubed tangent line L . Hence, p dim Q − (3µf + 1).p =0. That is, (1) 3µf +1 ∈ Gp (Q). The spaces Q − ℓ.p = φ for ℓ ≥ 3µf +2. Consequently, (1) 3µf +2 ∈ Gp (Q). Consequently, (1) Gp (Q)= {1, 2, 3, 1+ µf , 2+ µf , 3+ µf , 2µf +1, 2µf +2, 3µf +1, 3µf +2}. Finally, it is well known that curves of genus 10 can be embedded into algebraic curves of degree 6. Hence, by the famous Bezout’s theorem, the tangent line meets C at the flex point p with 3 ≤ µf ≤ 6. On the other hand, by Theorem 1.2, it follows that, µf =66 . Which completes the proof. Corollary 2.6. If a smooth projective curve X of g = 10 has 4-flex points, then X is hyperelliptic. Corollary 2.7. On a smooth non-hyperelliptics projective plane curve C of g = 10, the 1-weight of a flex point is given by (1) ωp (Q)=13µf − 37, where µf is the multiplicity of the tangent line Lp to C at p. Proof. Let p be a flex point on C, then by Theorem 2.5 (1) Gp (Q)= {1, 2, 3, 1+ µf , 2+ µf , 3+ µf , 2µf +1, 2µf +2, 3µf +1, 3µf +2}. 5 Consequently, (1) g ωp (Q) := Σr=0(nr − r) = (1+ µf − 4)+(2+ µf − 5)+(3+ µf − 6)+(2µf +1 − 7)+(2µf +2 − 8) + (3µf +1 − 9)+(3µf +2 − 10) = 13µf − 37. (q) Notation. Let Fi (C) be the set of i-flex points which are q-Weierstrass points on (q) (q) C. Also, let NFi (C) denotes the cardinality of Fi (C). Corollary 2.8. For a smooth non-hyperelliptic projective plane curve C of g = 10, (1) the maximal cardinality of Fi is given by the following table: (1) i Maximum NFi (C) 1 495 2 66 3 35 4 0 Proof. The number of the 1-Weierstrass points on C is 990 counted with their 1- weight. Hence, 990 NF (1) ≤ [ ], i 13(i + 2) − 37 where i =1, 2, 3.
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